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We present a new analytical description of cosmic structure formation in a mixture of dark and baryonic matter, using the framework of Kinetic Field Theory (KFT) -- a statistical field theory for classical particle dynamics. So far, KFT has only been able to describe a single type of particles, sufficient to consider structure growth due to the gravitational interactions between dark matter. However, the influence of baryonic gas dynamics becomes increasingly relevant when describing smaller scales. In this paper, we thus demonstrate how to extend the KFT formalism as well as a previously presented resummation scheme towards describing such mixtures of two particle species. Thereby, the gas dynamics of baryons are accounted for using the recently developed model of Mesoscopic Particle Hydrodynamics. Assuming a flat $Λ$CDM Universe and a simplified model for the thermal gas evolution, we demonstrate the validity of this approach by computing the linear evolution of the individual and total matter power spectra between the epoch of recombination and today. Our results correctly reproduce the expected behaviour, showing a suppression of both baryonic and dark matter structure growth on scales smaller than the baryonic Jeans length, in good agreement with results from the numerical Boltzmann solver CLASS. Nonlinear corrections within this approach will be investigated in upcoming works.